24-Pet-A3 Fundamental Reservoir Engineering · December 2014
Question 5 of 7: Capillary Pressure — Water-Oil Contact and Water-Oil Ratio
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
98-Pet-A3 — Fundamental Reservoir Engineering · National Exams, December 2014 · 3 hours, closed book, Casio/Sharp approved calculator only · five (5) questions constitute a complete exam paper (the first five as they appear in the answer book are marked), all questions equal value, all parts of a multipart question equal weight.
Reference texts: Ahmed, T., Reservoir Engineering Handbook, 5th ed. (Darcy's law and fluid potential, transient well testing, p/Z and oil material balance, capillary pressure/relative permeability); Craft, B.C. & Hawkins, M.F., Applied Petroleum Reservoir Engineering, 3rd ed. (reservoir drive mechanisms, material balance fundamentals); Lyons, W.C. (ed.), Standard Handbook of Petroleum and Natural Gas Engineering, 3rd ed. (Standing–Katz Z-factor correlation); McCain, W.D., The Properties of Petroleum Fluids, 3rd ed. (capillary pressure and relative permeability laboratory data).
Question 5: Capillary Pressure — Water-Oil Contact and Water-Oil Ratio (20 marks)
$k_{ro}=1$ at $S_w=0.2$; $k_{ro}=0$ at $S_w=0.82$; $k_{rw}=0.45$ at $S_w=1$; crossover $k_{ro}=k_{rw}=0.18$ at $S_w=0.64$
Find. (a) Depth of the WOC; (b) approximate WOR for a well completed 36 ft above the FWL.
Fig. 4 — Capillary pressure vs. water saturation. The curve does not reach $P_c=0$ by $S_w=1$; it flattens at $\approx3$ psi, which is what separates the WOC from the (deeper) FWL.
Fig. 5 — Relative permeability vs. water saturation (Corey-type fit through the stated anchor points), used to read $k_{ro}$, $k_{rw}$ at $S_w=0.4$.
Approach. Convert the capillary-pressure curve into height above the FWL via the buoyancy relation $h=144P_c/(\rho_w-\rho_o)$; the WOC is the depth at which the curve's own data show $S_w$ first reaching 100%, which here happens at a finite $P_c\approx3$ psi, not at $P_c=0$ (the FWL). For part (b), convert the 36 ft completion height to $P_c$, read $S_w$ off the same curve, then read $k_{ro},k_{rw}$ at that saturation off the relative-permeability curve (fit as a Corey power law through its stated anchor points) and combine with the PVT/viscosity data via the surface WOR relation.
WOC location. $h=\dfrac{144P_c}{\rho_w-\rho_o}=\dfrac{144P_c}{20}=7.2P_c$ ft above the FWL. The curve's own asymptotic value at $S_w=1$ (100% water) is $P_c\approx3$ psi, so $h=7.2(3)=21.6$ ft. Since 100% water is reached at a small positive $P_c$ (not at $P_c=0$, which defines the deeper FWL), the WOC sits 21.6 ft shallower than the FWL: $\boxed{\text{WOC}=3000-21.6=2978.4\ \text{ft subsea}}$.
Saturation at the completion. 36 ft above the FWL, $P_c=\dfrac{(\rho_w-\rho_o)(36)}{144}=\dfrac{20(36)}{144}=5.0$ psi, which is exactly the curve's own labelled point $\boxed{S_w=0.4}$.
Relative permeabilities at $S_w=0.4$. Fitting Corey exponents through the stated anchors ($S_{wi}=0.2$, $S_{or}=0.82$ for $k_{ro}$; full range for $k_{rw}$) so that both curves pass through the crossover ($S_w=0.64$, $k_{ro}=k_{rw}=0.18$) gives $n_o=1.39$, $n_w=1.53$, and at $S_w=0.4$: $\boxed{k_{ro}=0.583}$, $\boxed{k_{rw}=0.054}$.
Check: the source supplies the capillary-pressure and relative-permeability curves graphically, with the specific labelled points quoted above; $k_{ro}(0.4)$ and $k_{rw}(0.4)$ are obtained by fitting a Corey-type power law through the curves' own stated anchor points (endpoints plus the stated crossover) rather than by reading an unlabelled position off the chart, since those three points are the only values the source itself calls out numerically.
